11x11 Queens Puzzle #164
Difficulty: easy. Place 11 queens on this 11x11 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 24-step reasoning is written out below.
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Step-by-step solution for puzzle #164
What follows is a complete solve of puzzle #164, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 11 queen placements and 13 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the proof by contradiction.
Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 3, column 11 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 11, the rest of the blue region and every square adjacent to it — diagonals included.
Step 2. The indigo region is down to one open square, at row 1, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 10, the remaining indigo squares, and the ring of neighbours around row 1, column 10.
Step 3. The mint region has 2 squares left, spread across rows 2 and 4, columns 1 and 2. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 4. Wherever red ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Pale yellow is down to 3 candidates in rows 7 and 8, columns 2 and 3, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
The board after step 5 — 2 of 11 queens placed. Step 6. The orange region has 4 squares left, spread across rows 9–11, columns 8 and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 7. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 8 squares in all.
Step 8. Here is the pairing argument. The red, orange, and teal regions have no open squares outside columns 7–9 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 12 squares come off in one stroke.
Step 9. Test row 2, column 3 for amber: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 10. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 5. Follow the forced moves that would create and row 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 11. Test row 2, column 6 for amber: place a queen there, run the forced consequences through, and the slate region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 11 — 2 of 11 queens placed. Step 12. Now a short proof by contradiction. Suppose the amber queen went to row 4, column 4. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Test row 4, column 6 for amber: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 5, column 4. Follow the forced moves that would create and row 7 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Test row 5, column 6 for amber: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 16. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 4; mint has one square left, so its queen takes row 4, column 2; red has one square left, so its queen takes row 6, column 9; pale yellow has one square left, so its queen takes row 7, column 3; orange has one square left, so its queen takes row 9, column 8; teal has one square left, so its queen takes row 11, column 7; row 5 is down to one square, putting a queen on row 5, column 5; khaki has one square left, so its queen takes row 8, column 6; sea green has one square left, so its queen takes row 10, column 1. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 10, column 1.
The board after step 16 — 11 of 11 queens placed.
That completes the board: 11 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.
About 11x11 Queens puzzle #164
This is puzzle number 164 in the 11x11 collection, rated easy. The grid holds 121 squares split between eleven colored regions, and a finished board carries eleven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #164 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #164 are: blue (1 square), pale yellow (3 squares), teal (3 squares), mint (4 squares), indigo (7 squares), orange (9 squares), sea green (12 squares), red (15 squares), khaki (17 squares), slate (24 squares), amber (26 squares). An even split would hand every color 11 squares, so the blue region is running 10 squares under that average while the amber region is the roomiest area on the grid with 26 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 1 square while amber has 26 to spare, a gap of 25. Attack the cramped end and the roomy regions tend to resolve themselves.
A region's outline can constrain it more than its size does. Here blue is the most solid, filling 100% of its 1×1 box, and khaki the most diffuse at 49% of its 5×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #164, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Blue occupies 1 square within rows 3–3, columns 11–11. It never leaves row 3, so its queen is locked to that row.
Pale yellow: 3 squares across rows 7–8, columns 2–3.
Teal covers 3 squares, spanning 2 rows and 2 columns (rows 10–11, columns 7–8).
Mint occupies 4 squares within rows 2–4, columns 1–2.
Indigo: 7 squares across rows 1–3, columns 8–11.
Orange covers 9 squares, spanning 3 rows and 4 columns (rows 9–11, columns 8–11).
Sea green occupies 12 squares within rows 6–11, columns 1–4.
Red: 15 squares across rows 4–9, columns 7–11.
Khaki covers 17 squares, spanning 5 rows and 7 columns (rows 7–11, columns 3–9).
Slate occupies 24 squares within rows 4–8, columns 1–9.
Amber: 26 squares across rows 1–5, columns 1–9.
Where to start on this board
Open on the tightest color. Here that is blue, confined to 1 square between rows 3 and 3 and columns 11 and 11. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #164.
Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. Then look at which colors just lost ground: on a grid this size that one observation usually forces a second region within a move or two.
Work outward to the other flat regions next. Pale yellow and teal are no more than two rows tall, so their queens are confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.
The tightest lines on puzzle #164
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 5, 6 and 10 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 8 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
Two checks, run alternately, will carry you through the 121 squares of this grid. First: has any of the eleven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 11x11 boards almost always run one of those checks and forget the other.
Resist the plausible-looking queen. On #164 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #164 is rated easy
The rating reflects how much the board concedes up front. Puzzle #164 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 25 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
Techniques that work well at 11x11
At 11x11 you are managing 121 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 7 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
- There is always exactly one valid solution. Use logic, never guessing, to find it.
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Questions about this puzzle
How hard is 11x11 Queens puzzle #164?
Puzzle #164 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #164 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #164, has been verified to have exactly one valid arrangement of eleven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #164?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.